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| E5 Extra Class Electrical Principles..T5C Impedance Calculations, Series Circuits, Polar-Coordinates | |||||||||||||
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Introduction The following tutorial explains step-by step how to solve series impedance calculations. A calculator is required, since trigonometric functions will be used. A knowledge of Trig. is helpful but not required. A knowledge of the symbols Z, ø L, C, and X.
Series XL + R Example Problem: We have a circuit with a 100 Ω resistor and a 1mH inductor in Series at 15.916kHz. Find the circuits impedance.Step [1] Find the inductor's inductive reactance. XL= 2πfL or.. 2π * (15.916 * 103) * (10-3H) Answer = 100 Ohms at 15.916kHz. Step [2] Draw a Diagram To solve the problem lets review what we know: The circuit is in series, with a resistor value of 100Ohms at the frequency of operation (15.916kHz). To solve the problem, we construct a coordinate graph. Grab a sheet of paper and follow along. Draw two lines perpendicular to each other. Label the horizontal axis as 'R', and the vertical axis as 'X'. The Y-axis is 'X' for Reactance of the circuit. X-axis as 'R' for the pure resistance of the circuit. To solve problems with this coordinate plane graph, we plot any values of reactance in the vertical (Y) plane. They would be lines drawn straight up an down. Any resistance values would be drawn on the horizontal plane (X), and would be lines drawn side-to-side. 
The pure resistance(R) of the circuit is 100 ohms. Draw an arrow on the R-axis and label it 100 ohms. Likewise, the Reactance(X) of the circuit is inductive. If the reactance is inductive the arrow for the XL is show pointing up 100 Ohms from the R-axis. This is important because the XL of a circuit is always drawn pointing up from the R-axis. On the otherhand the XC of a circuit is always drawn pointing down. This may be confusing so don't worry, just remember those 2 facts. The arrow completing the triangle-shape, 'Z', is the impedance. The length of this arrow in Ohms is the value that we are looking to solve. Basically we have a triangle with 2 sides with a length of 100. How can we solve for the unknown side Z? Step [3] Use Trig The side we are trying to solve for is called the hypotenuse in trig. We can solve for the hypotenuse by using trig functions. To solve for Z we can simply use the pythagorean them. |Z|= Square root of (R2 + XL2) |Z| = square root of 1002 + 1002. = 141 Ω. Now we know the impedance of the circuit. Next we will solve for the phase angle of the circuit. Phase Angles
The phase angle lets us know the amount of phase shift a signal goes through when it passes through this particular circuit. To solve for this we need to look at your calculator and make sure you have the following buttons: SIN-1 COS-1 and TAN-1. You may have to press the 2nd button and then the SIN button to get the SIN-1 to work on your calculator. To solve for the angle, inside a triangle with only the length of the sides to work with we use the inverse trig function. In this case we can use the TAN-1 function to solve for θ. TAN-1θ = opp/adj Why Tan, and not another Trig function? Because we need to look at the information we have been given (from previous problem), and use a trig function that uses those sides that we know. The Opposite and the Adjacent sides have been given to us (100) and we can use those 2 numbers together with the TAN-1 function to solve for theta (the angle), which is the answer we are looking for. TAN-1 θ = (100/100) = tan-1(1). θ = TAN-1(1) = 45 degrees. The phase angle (θ) = 45 degrees. Polar Coordinate Form Step [5] State the answer in polar coordinate form.This is easy. Just write 141Ω /45degrees. the / is a symbol for an angle. Next time, we will discuss the R+XC Series Circuit. Stay tuned.
1.) Solve for the reactance of the inductor (L). 2.) Draw a rectangular coordinate graph, and plot the triangle. XL is positive, and XC is negataive. (And always lies below the R-axis.) 3.) Solve for the impedance by using the pythagorean theorem. 4.) Solve for the phase angle using one of the trig functions. 5.) State the answer in polar coordinate form.
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